Question

A) The reference desk of a college library receives requests for assistance. Assume that a Poisson probability distribution with an arrival rate of 10 requests per hour can be used to describe the arrival pattern and that service times follow an exponential probability distribution with a service rate of 12 requests per hour. Use this information to determine the following operating characteristics for the system:

1) The probability that no customers are in the system

2) The average number of customers waiting

3) The average number of customers in the system

4) The average time a customer spends waiting

5) The average time a customer spends in the system

6) The probability that arriving customers will have to wait for service

B) Movies Tonight is a typical video and DVD movie rental outlet for home-viewing customers. During the weeknight evenings, customers arrive at the store with an arrival rate of 1.25 customers per minute. The checkout clerk has a service rate of 2 customers per minute. Assuming Poisson arrivals and exponential service times, determine the following operating characteristics for the system:

1) The probability that no customers are in the system

2) The average number of customers waiting

3) The average number of customers in the system

4) The average time a customer spends waiting

5) The probability that arriving customers will have to wait for service, and also answer

6) Do the operating characteristics indicate that the one-clerk checkout system provides an acceptable level of service?

Answer 1

As per the Chegg answering guidelines for multiple unrelated questions, answering the first question only

The arrival rate is 10 per hour while the service rate is 12 per hour

1) The probability that no customers are in the system, P0

2) The average number of customers waiting, Lq

3) The average number of customers in the system, L

4) The average time a customer spends waiting, Wq

5) The average time a customer spends in the system, W

6) The probability that arriving customers will have to wait for service

The probability that arriving customers will have to wait is the same probability that there is atleast 1 customer in the system. Now the probability that there is atleast 1 customer in the system is 1 minus probability that there is no customer in the system,P0. As caclulated in Q1, P0 is 0.1667. Hence the probability that there is atleast 1 customer in the system is 1-0.1667 = 0.8333. Hence the probability that the arriving customers will have to wait for the service is 0.8333